Critical Angular Velocity Research Articles

The Melnikov criterion of instability for random rocking dynamics of a rigid block with an attached secondary structure

In this paper we investigate the effect of structural flexibility on rocking motion of a system consisting of a free standing rigid block with an attached chain of uniaxially moving point masses. Motion is excited by random acceleration of the ground; instability is associated with overturning of the overall structure. The condition of instability is constructed by the stochastic Melnikov method. We demonstrate a twofold effect of structural flexibility on the rocking response. The attached structure may increase the critical angular displacement and velocity in comparison with the similar parameters of the single rigid block. At the same time, the enlargement of the domain of stability enhances the contribution of the random perturbation in the Melnikov process. As a result, a lower level of random forcing can result in overturning of the structure. As an example, an effect of a single-mass secondary structure on the dynamic behavior of the system is discussed. The paper is restricted to the consideration of seismic vulnerability of the structure. A similar approach can be applied to systems with wind or wave excitation.

Nonaxisymmetric Shapes of a Magnetically Levitated and Spinning Water Droplet

The shape of a weightless spinning liquid droplet is governed by the balance between the surface tension and centrifugal forces. The axisymmetric shape for slow rotation becomes unstable to a nonaxisymmetric distortion above a critical angular velocity, beyond which the droplet progresses through a series of 2-lobed shapes. Theory predicts the existence of a family of 3- and 4-lobed equilibrium shapes at higher angular velocity. We investigate the formation of a triangular-shaped magnetically levitated water droplet, driven to rotate by the Lorentz force on an ionic current within the droplet. We also study equatorial traveling waves which give the droplet threefold, fourfold, and fivefold symmetry.

Numerical Simulations on Stationary States for Rotating Two-Component Bose-Einstein Condensates

We numerically study stationary states such as ground, symmetric and central vortex states as well as their energy diagrams for rotating two-component Bose-Einstein condensates (BECs), which are stationary solutions of time-independent coupled Gross-Pitaevskii equations with an angular momentum rotational term. We compute these stationary states by using normalized gradient flows with a backward Euler finite difference discretization, which is proved to be efficient and accurate. By using this discretization, we find various ground state configurations with several vortices for the two-component BECs, which are novel but not found in rotating two-component BECs; we also find that the critical angular velocity at which the ground states lose symmetry.

Stability analysis of a substructured model of the rotating beam

One of the most well-known situations in which nonlinear effects must be taken into account to obtain realistic results is the rotating beam problem. This problem has been extensively studied in the literature and has even become a benchmark problem for the validation of nonlinear formulations. Among other approaches, the substructuring technique was proven to be a valid strategy to account for this problem. Later, the similarities between the absolute nodal coordinate formulation and the substructuring technique were demonstrated. At the same time, it was found the existence of a critical angular velocity, beyond which the system becomes unstable that was dependent on the number of substructures. Since the dependence of the critical velocity was not so far clear, this paper tries to shed some light on it. Moreover, previous studies were focused on a constant angular velocity analysis where the effects of Coriolis forces were neglected. In this paper, the influence of the Coriolis force term is not neglected. The influence of the reference conditions of the element frame are also investigated in this paper.

Skyrme Strings

We construct nontopological string solutions with U (1) Noether charge in the Skyrme model with a pion mass term, and examine their stability by taking linear perturbations. The solution exhibits a critical angular velocity beyond which the configuration energetically prefers to decay by emitting pions. This critical point is observed as a cusp in the relation between energy and charge. We find that the maximum length for the string to be stable is comparable to the size of one skyrmion. Beyond the length, it is unstable to decay. This instability raises the possiblity of dynamical realization of Skyrme strings from monopole strings inside a domain wall.

Destroying Superfluidity by Rotating a Fermi Gas at Unitarity

We study the effect of the rotation on a harmonically trapped Fermi gas at zero temperature under the assumption that vortices are not formed. We show that at unitarity the rotation produces a phase separation between a nonrotating superfluid (S) core and a rigidly rotating normal (N) gas. The interface between the two phases is characterized by a density discontinuity n(N)/n(S)=0.85, independent of the angular velocity. The depletion of the superfluid and the angular momentum of the rotating configuration are calculated as a function of the angular velocity. The conditions of stability are also discussed and the critical angular velocity for the onset of a spontaneous quadrupole deformation of the interface is evaluated.

Convective envelopes in rotating OB stars

Aims. We study the effects of rotation on the outer convective zones of massive stars. Methods. We examine the effects of rotation on the thermal gradient and on the Solberg-Hoiland term by analytical developments and by numerical models. Results. Writing the criterion for convection in rotating envelopes, we show that the effects of rotation on the thermal gradient are much larger and of opposite sign to the effect of the Solberg-Hoiland criterion. On the whole, rotation favors convection in stellar envelopes at the equator and to a smaller extent at the poles. In a rotating 20 $M_{\odot}$ star at 94% of the critical angular velocity, there are two convective envelopes, with the bigger one having a thickness of 13.2% of the equatorial radius. In the non-rotating model, the corresponding convective zone has a thickness of only 4.6% of the radius. The occurrence of outer convection in massive stars has many consequences.

Formation of Terrestrial Planets from Protoplanets. II. Statistics of Planetary Spin

The final stage of terrestrial planet formation is known as the giant impact stage, where protoplanets collide with one another to form planets. The initial spin state of terrestrial planets is determined at this stage. We statistically investigate the spin parameters of terrestrial planets assembled from protoplanets using N-body simulations. As initial conditions, we adopt the oligarchic growth model of protoplanets. For the standard disk model, typically two Earth-sized planets form in the terrestrial planet region. We find that the spin angular velocity of the planets is well expressed by a Gaussian distribution, and their obliquity is well expressed by an isotropic distribution. The typical spin angular velocity is given by the critical spin angular velocity for rotational instability under the assumption of perfect accretion in collisions. We show the dependencies of the spin parameters on the initial protoplanet system parameters. The initial orbital separation and velocity anisotropy of protoplanets barely affect the spin parameters. The bulk density of protoplanets does not affect the obliquity distribution, while the spin angular velocity increases with the bulk density.

EFFECTS OF CONSTRAINTS ON BIPEDAL BALANCE CONTROL DURING STANDING

In this work, the effects of the gravity constraint, friction constraint, center of pressure (COP) constraint, and tip-over constraint on balance control of a simplified two-dimensional biped during standing are investigated. The bounds imposed on the control torque due to the constraints are first determined. Such control bounds have significant effects on designing balance control laws and can be used to predict the type of falls if the constraint is violated. It is found that there exists a critical angular velocity, above which, regardless of the control torque, the constraints will be violated. It is also found that the gravity constraint is the least important since it is always satisfied when the friction constraint is satisfied. For the biped under study, the friction constraint determines the critical angular velocity in most of the region around the upright position, while the COP constraint dictates the bounds of the control torque. Since the tip-over constraint is equivalent to the COP constraint as both feet are on the same level ground, it is concluded that the tip-over constraint is the most important when the magnitude of the angular velocity is below the critical value.

Vortices in rotating Bose–Einstein condensates confined in homogeneous traps

We investigate analytically the thermodynamical stability of vortices in the ground state of rotating 2D Bose–Einstein condensates confined in asymptotically homogeneous trapping potentials in the Thomas–Fermi regime. Our starting point is the Gross–Pitaevskii energy functional in the rotating frame. By estimating lower and upper bounds for this energy, we show that the leading order in energy and density can be described by the corresponding Thomas–Fermi quantities and we derive the next order contributions due to vortices. As an application, we consider a general potential of the form V(x,y)=(x2+λ2y2)s/2 with slope s∈[2,∞) and anisotropy λ∈(0,1] which includes the harmonic (s=2) and ‘flat’ (s→∞) traps, respectively. For this potential, we derive the critical angular velocities for the existence of vortices and show that all vortices are single-quantized. Moreover, we derive relations which determine the distribution of the vortices in the condensate i.e. the vortex pattern.

Ground State and Single Vortex for Bose–EinsteinCondensates in Anisotropic Traps

For Bose–Einstein condensation of neutral atoms in anisotropictraps at zero temperature, we present simple analytical methods forcomputing the properties of ground state and single vortex ofBose–Einstein condensates, and compare those results to extensivenumerical simulations. The critical angular velocity for productionof vortices is calculated for both positive and negative scatteringlengths a, and find an analytical expression for the large-N limitof the vortex critical angular velocity for a>0, and the criticalnumber for condensate population approaches the point of collapsefor a<0, by using approximate variational method.

Internal gravity wave resonance of thermal convection fields in rectangular cavities with heat‐flux vibration (effects of aspect ratio on the fields)

AbstractIn this paper the thermal convection field and its resonance phenomena in a rectangular cavity with heat‐flux vibration are numerically examined and the results are compared with those of a square cavity. As in the case of α=1, the critical angular velocity at which the relative amplitude of the midplane Nusselt number αm has a local maximum agrees very well with the resonance angular velocity of the internal gravity wave ωr, estimated by the theoretical equation proposed by Thorpe, even when the aspect ratio is α=5 and the Prandtl number is Pr=0.71 for a range of the Rayleigh number Ra. However, αm has two local maxima for a larger Ra, which is peculiar to the case of larger α. The time variation of sub‐components of the fluctuating component of the midplane Nusselt number shows that the phase at the maximum value of αm agrees well with that of the sub‐component of velocity for the first resonance angular velocity ωr. For the other angular velocity ωr2, the phase of αm agrees with that of the sub‐component of temperature. Moreover, we found that the boundary angular velocity ω0 between the first two of the five ω regions, which classify the thermal convection fields against ω, can be expressed by a function of α, Ra, and Pr and that αm is independent of α and Ra for a relatively wide range of ω/ω0. © 2007 Wiley Periodicals, Inc. Heat Trans Asian Res, 36(3): 158– 171, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/htj.20149

Rotation of the leaky dielectric particle in a rotating electric field

We study the rotation of a weakly conducting particle around its axis of symmetry under the action of the external electric field, which spins in the plane normal to the axis of symmetry of the particle. The particle is embedded in a homogeneous stationary medium with finite electric conductivity permittivity that are different from the corresponding parameters of the particle. We determined the dependence of the particle angular velocity upon the amplitude angular velocity of the electric field. It is shown that depending upon the ratios of the particle electric conductivity permittivity to the corresponding parameters of the host medium the direction of rotation of the particle can be identical or opposite to the direction of rotation of the external electric field. We determined the amplitude-dependent critical angular velocity of the external electric field that separates the domains with two possible regimes of rotation of the particle. In the first domain the particle rotates only in one direction, while in the second domain the particle may rotate in two directions. We investigated also the stability of different regimes of rotation of the particle.

A renormalized Gross–Pitaevskii theory and vortices in a strongly interacting Bose gas

We consider a strongly interacting Bose–Einstein condensate in a spherical harmonic trap. The system is treated by applying a slave-boson representation for hardcore bosons. A renormalized Gross–Pitaevskii theory is derived for the condensate wavefunction that describes the dilute regime (like the conventional Gross–Pitaevskii theory) as well as the dense regime. We calculate the condensate density of a rotating condensate for both the vortex-free condensate and the condensate with a single vortex and determine the critical angular velocity for the formation of a stable vortex in a rotating trap.

A solution to the ground and single vortex states of Bose-condensed gas in an axially symmetric harmonic trap

For a Bose-condensed gas in an axially symmetric harmonic trap at zero temperature, a new trial wave function of the ground and single vortex states is considered based on the mean-field energy functional and variational method. Once the wave function is known, we can explore the particle-number dependence of relevant quantities, such as energy, chemical potential, aspect ratio of the velocity distribution and critical angular velocity for production of vortices. These results agree well with previous numerical solutions of the G-P equation given by Dalfovo et al.

Time-dependent constitutive modeling of drive belts – I. The effect of geometry and number of loading cycles

This paper presents constitutive modeling of dynamically loaded elastomeric products such as power transmission belts. During the normal operation of the belts certain segments of the belt structure are loaded with a tooth-like periodic (cyclic) loading. When the time-dependent properties of the elastomeric material “match” the time-scale of the dynamic loading a strain accumulation process occurs. The critical angular velocity is proportional to the ratio of the belt length to the common diameters of the pulleys. The magnitude of the strain accumulated in each loading cycle decreases with an increase in belt length. For a given belt geometry the critical angular velocity increases with the number of loading cycles. At the same time the magnitude of the accumulated strain decreases non-linearly as the number of loading cycles increases. However if the belt operates at or in the close vicinity of its critical angular velocity it will almost certainly fail!

Research on the Coning Motion of Wrap-Around Fin Projectiles

In this paper, a new coordinate frame is defined, and the mathematical model of the coning motion of wrap-around-fin projectiles is set up. The dynamic stability of the coning motion is investigated by means of Lyapunov First Theory and the centre manifold theory. The influences of roll angular velocity and the free-stream Mach number of the projectile on the coning motion are analyzed theoretically and numerically. The results indicate the existence of coning motion and stability bifurcation at a critical roll angular velocity.

Magneto-rotational instability in differentially rotating liquid metals

We study the stability of Couette flow between two cylinders in the presence of axial magnetic field in local WKB approximation. We find the analytical expression of the critical angular velocity minimized over the wave number and the imposed magnetic field as a function of the measure of deviation of the rotation law from the Rayleigh line. The result found is in a good agreement with the previously known numerical results based on the global analysis. We perform a minimization of the critical Reynolds number over the wave number at fixed magnetic field both analytically and numerically. We show that a compromise between resistive suppression of magneto-rotational instability at weak magnetic field and the increase of the critical Reynolds number with the increase of magnetic field is possible. It takes place at moderate values of magnetic field of order 3 × 10 2 gauss giving the critical Reynolds number of order 4 × 10 4 .

Internal Gravity Wave Resonance of Thermal Convection Fields in Rectangular Cavities with Heat-Flux Vibration (Effects of Aspect Ratio on the Fields)

In this paper the thermal convection field and its resonance phenomena in a rectangular cavity with heat-flux vibration were numerically examined and the results were compared with the ones of square cavity. In the case of the aspect ratio α=5 and the Prandtl number Pr=0.71, for a range of the Rayleigh number Ra the critical angular velocity ωc, at which the relative amplitude of midplane Nusselt number αm has a local maximum, agrees very well with the resonance angular velocity of internal gravity wave ωγ, estimated by the theoretical equation proposed by Thorpe, as observed for α=1. However αm has two local maxima for larger Ra, which is peculiar to the case of larger αm The time variation of components of fluctuating midplane Nusselt number shows that the phase at the maximum value of αm agrees well with that of the fluctuating component of velocity for the first resonance angular velocity ωγ. For the other angular velocity ωγ2 the phase of αm agrees with that of the fluctuating component of temperature. Moreover, we found that the boundary angular velocity ω0 between the first two ω regions among five regions, which classify the thermal convection fields against ω, can be expressed by a function of a, Rα and Pr and that αmis independent of α and Rα for relatively wide range of ω/ω0.

Superfluid turbulence in rotating containers: Phenomenological description of the influence of the wall

In this paper a previous equation for the evolution of vortex line density L in counterflow superfluid turbulence in rotating containers is generalized, in order to take into account the influence of the walls. This model incorporates the effects of counterflow velocity V and of angular velocity {omega} of the container, and introduces corrective terms depending on {delta}/d, {delta} being the intervortex spacing, of the order L{sup -1/2}, and d the diameter of the channel. The stability of the solutions for L, for several regimes of averaged counterflow velocity V and angular velocity {omega}, is analyzed. Our mathematical analysis reveals that qualitative consistency allows us to reduce the four coefficients characterizing the dependence on {delta}/d to only one additional independent coefficient, linked to the critical angular velocity {omega}{sub c} needed for the appearance of vortex lines in a rotating superfluid.